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Email letter I hope you are doing fine. After contacted with Professor Muller. Head of department about apply for 3 months fellowship training in your institution and after send all requirements he requested. He ask me to ask you about which next avaiable dated for fellowship post

Corrections · 5

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Email letter

I hope you are doing fine. After I contacted professor Muller, the head of the department, about applying for a 3 month fellowship training in your institution and sending all the requirements he requested, he asked me to ask you about the next available date for a Post-graduate fellowship.
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Email letter

I hope you are doing fine. After contacted talking to with Professor Muller, head of department, about applying for a 3-months fellowship training in your institution, and after sending all requirements he requested, he asked me to ask you about which the next available dated for the fellowship post position.

Editor's comment: I recommend reading about basic grammar rules for gerunds and infinitives.
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The situation here is not clear. Is the writer the person who contacted Professor Muller? Is the writer applying for the fellowship? who is "he" in "all requirements he requested" who is "he" AND who is 'you' in "he asked me to ask you"

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